Objects in 2 dimensions

Reading the February edition of the New Scientist (about space-time being possiblly quantized in its own right) I read that the event horizon of a black hole is a 2-dimensional entity that may possibly encode all the information to describe the 3- 0r ->3 dimensional universe inside.

That is interesting of course but am I right to wonder how the surface of the black hole can be described as 2-dimensional in the first place?

In my poor little mind it would only qualify as 2 dimensional (and then only in theory) if it was purely idealised as a surface with zero width.

This would be impossible unless the black hole was to exist in isolation to the rest of the universe.

To my mind 1- ,2- 3- dimensional obnjects are all idealisations from the established 4- or higher dimensional setup we work in at the moment.

That seems to be my main point: I can cope with gazillions of hypothetical extra dimensions but not with any subtraction of those we already seem to be dealing with.

Have I got things by the wrong handle somehow or am I just naturally obtuse (or both obviously) ?

When it comes to dimensions, you have to think a little differently. When you hear of a two dimensional object you're probably thinking of a infinitely thin flat sheet of paper. It has no depth, just length and width.

However, in modern math and physics the following objects are all classified as two dimensional: an infinite cylinder, the shell of a sphere, the shell of a torus, and the mobius strip. Thats because if you zoomed in on each object, the surface would look like R^2, or the flat plane. If something lived on these spaces they would only know of 2 dimensions, since the tangent space to each of these objects is 2 dimensional.

FYI, a referred to the shell of a sphere and torus to emphasize that they are hollow. Normally a shell of a 3 dimensional ball is referred to as a 2d dimensional sphere and the shell of a doughnut is just called a torus.

So the surface of a black hole is a 2d sphere according to these definitions. How is it possible to encode information on a smaller dimensional object? Well that's because gravity is weird :).

However, in modern math and physics the following objects are all classified as two dimensional: an infinite cylinder, the shell of a sphere, the shell of a torus, and the mobius strip. Thats because if you zoomed in on each object, the surface would look like R^2, or the flat plane. If something lived on these spaces they would only know of 2 dimensions, since the tangent space to each of these objects is 2 dimensional.

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thanks but

how is it possible to zoom in since zooming in requires another dimension?

Whilst the zooming continues the extra (3rd and 4th or more) dimension is always implicit.

The completion of the zooming never actually occurs and neither is it possible to put yourself in the place of an inhabitant except in an idealised sense.

I mean aren't all these surfaces just mathematical abstractions (I appreciate that Idealism is an accepted philosophical discipline but I thought Materialism was more de rigeur these days)

When it comes to dimensions, you have to think a little differently. When you hear of a two dimensional object you're probably thinking of a infinitely thin flat sheet of paper. It has no depth, just length and width.

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More nitpicking: an infinitely thin flat piece of paper is also an idealism.In my mind it is 3- dimensional no matter how thin it is.

At no point does it become 2 dimensional except as a mathematical abstraction.